{"status":"ok","message-type":"work","message-version":"1.0.0","message":{"indexed":{"date-parts":[[2025,10,12]],"date-time":"2025-10-12T04:46:39Z","timestamp":1760244399669,"version":"build-2065373602"},"reference-count":18,"publisher":"MDPI AG","issue":"12","license":[{"start":{"date-parts":[[2022,12,2]],"date-time":"2022-12-02T00:00:00Z","timestamp":1669939200000},"content-version":"vor","delay-in-days":0,"URL":"https:\/\/creativecommons.org\/licenses\/by\/4.0\/"}],"funder":[{"DOI":"10.13039\/501100004826","name":"Natural Science Foundation of Beijing","doi-asserted-by":"publisher","award":["1222012","2019YFB2006602","11801296"],"award-info":[{"award-number":["1222012","2019YFB2006602","11801296"]}],"id":[{"id":"10.13039\/501100004826","id-type":"DOI","asserted-by":"publisher"}]},{"name":"National Key Research and Development Project","award":["1222012","2019YFB2006602","11801296"],"award-info":[{"award-number":["1222012","2019YFB2006602","11801296"]}]},{"name":"Natural Science Funds of China","award":["1222012","2019YFB2006602","11801296"],"award-info":[{"award-number":["1222012","2019YFB2006602","11801296"]}]}],"content-domain":{"domain":[],"crossmark-restriction":false},"short-container-title":["Axioms"],"abstract":"<jats:p>In 2010, Vuki\u010devi\u0107 introduced an new graph invariant, the inverse sum indeg index of a graph, which has been studied due to its wide range of applications. Let Bnd be the class of bipartite graphs of order n and diameter d. In this paper, we mainly characterize the bipartite graphs in Bnd with the maximal inverse sum indeg index. Bipartite graphs with the largest, second-largest, and smallest inverse sum indeg indexes are also completely characterized.<\/jats:p>","DOI":"10.3390\/axioms11120691","type":"journal-article","created":{"date-parts":[[2022,12,5]],"date-time":"2022-12-05T06:29:09Z","timestamp":1670221749000},"page":"691","update-policy":"https:\/\/doi.org\/10.3390\/mdpi_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A Complete Characterization of Bipartite Graphs with Given Diameter in Terms of the Inverse Sum Indeg Index"],"prefix":"10.3390","volume":"11","author":[{"given":"Guifu","family":"Su","sequence":"first","affiliation":[{"name":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Guanbang","family":"Song","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Junfeng","family":"Du","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Weixing","family":"Yang","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Gang","family":"Rao","sequence":"additional","affiliation":[{"name":"College of Mathematics and Physics, Beijing University of Chemical Technology, Beijing 100029, China"}],"role":[{"role":"author","vocabulary":"crossref"}]},{"given":"Jun","family":"Yin","sequence":"additional","affiliation":[{"name":"School of Computer, Qinghai Normal University, Xining 810000, China"}],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"1968","published-online":{"date-parts":[[2022,12,2]]},"reference":[{"key":"ref_1","doi-asserted-by":"crossref","unstructured":"Bondy, J.A., and Murty, U.S.R. (2008). Lukovits, Graph Theory, GTM 244, Springer.","DOI":"10.1007\/978-1-84628-970-5"},{"key":"ref_2","first-page":"243","article-title":"Bond additive modeling 1. Adriatic indices","volume":"83","year":"2010","journal-title":"Croat. Chem. Acta."},{"key":"ref_3","first-page":"261","article-title":"Bond additive modeling 2. Mathematical properties of max-min rodeg index","volume":"83","year":"2010","journal-title":"Croat. Chem. Acta"},{"key":"ref_4","doi-asserted-by":"crossref","first-page":"202","DOI":"10.1016\/j.dam.2014.11.013","article-title":"On the inverse sum indeg index","volume":"184","author":"Sedlar","year":"2015","journal-title":"Discret. Appl. Math."},{"key":"ref_5","doi-asserted-by":"crossref","first-page":"185","DOI":"10.1016\/j.dam.2016.09.014","article-title":"Sharp bounds on the inverse sum indeg index","volume":"217","author":"Azari","year":"2017","journal-title":"Discret. Appl. Math."},{"key":"ref_6","doi-asserted-by":"crossref","first-page":"42","DOI":"10.1016\/j.ipl.2018.02.006","article-title":"Some results on the inverse sum indeg index of a graph","volume":"134","author":"An","year":"2018","journal-title":"Inform. Process. Lett."},{"key":"ref_7","doi-asserted-by":"crossref","first-page":"18500061","DOI":"10.1142\/S1793830918500064","article-title":"The inverse sum indeg index of graphs with some given parameters","volume":"10","author":"Chen","year":"2018","journal-title":"Discret. Math. Algorithms Appl."},{"key":"ref_8","doi-asserted-by":"crossref","first-page":"59","DOI":"10.5614\/ejgta.2020.8.1.5","article-title":"Some new upper bounds for the inverse Sum indeg index of graphs","volume":"8","author":"Ali","year":"2020","journal-title":"Electron. J. Graph Theory Appl."},{"key":"ref_9","doi-asserted-by":"crossref","first-page":"2050152","DOI":"10.1142\/S1793557120501521","article-title":"A short note on inverse sum indeg index of graphs","volume":"14","author":"Balachandran","year":"2021","journal-title":"Asian-Eur. J. Math."},{"key":"ref_10","doi-asserted-by":"crossref","first-page":"5561033","DOI":"10.1155\/2021\/5561033","article-title":"Sharp Bounds for the Inverse Sum Indeg Index of Graph Operations","volume":"2021","author":"Rani","year":"2021","journal-title":"Math. Probl. Eng."},{"key":"ref_11","first-page":"125731","article-title":"On connected graphs and trees with maximal inverse sum indeg index","volume":"392","author":"Chen","year":"2021","journal-title":"Appl. Math. Comput."},{"key":"ref_12","doi-asserted-by":"crossref","first-page":"551","DOI":"10.46793\/KgJMat2004.551G","article-title":"Lower bounds for inverse sum indeg index of graphs","volume":"44","author":"Gutman","year":"2020","journal-title":"Kragujev. J. Math."},{"key":"ref_13","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.dam.2018.10.041","article-title":"Linear and non-linear inequalities on the inverse sum index","volume":"258","author":"Gutman","year":"2019","journal-title":"Discret. Appl. Math."},{"key":"ref_14","doi-asserted-by":"crossref","first-page":"123","DOI":"10.1016\/j.dam.2021.06.011","article-title":"A note on the minimum inverse sum indeg index of cacti","volume":"30","author":"Jiang","year":"2021","journal-title":"Discret. Appl. Math."},{"key":"ref_15","first-page":"29","article-title":"A note on chemical trees with maximal inverse sum indeg index","volume":"86","author":"Jiang","year":"2021","journal-title":"MATCH Commun. Math. Comput. Chem."},{"key":"ref_16","doi-asserted-by":"crossref","first-page":"252","DOI":"10.1007\/s40314-022-01964-6","article-title":"On two conjectures concerning trees with maximal inverse sum indeg index","volume":"41","author":"Lin","year":"2022","journal-title":"Comput. Appl. Math."},{"key":"ref_17","doi-asserted-by":"crossref","first-page":"131","DOI":"10.1016\/j.aml.2010.08.032","article-title":"Sharp upper bounds for Zagreb indices of bipartite graphs with a given diameter","volume":"24","author":"Li","year":"2011","journal-title":"Appl. Math. Lett."},{"key":"ref_18","doi-asserted-by":"crossref","first-page":"1165","DOI":"10.1016\/j.laa.2008.10.011","article-title":"On the spectral radius of bipartite graphs with given diameter","volume":"430","author":"Zhai","year":"2009","journal-title":"Linear Algebra Appl."}],"container-title":["Axioms"],"original-title":[],"language":"en","link":[{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/691\/pdf","content-type":"unspecified","content-version":"vor","intended-application":"similarity-checking"}],"deposited":{"date-parts":[[2025,10,11]],"date-time":"2025-10-11T01:33:07Z","timestamp":1760146387000},"score":1,"resource":{"primary":{"URL":"https:\/\/www.mdpi.com\/2075-1680\/11\/12\/691"}},"subtitle":[],"short-title":[],"issued":{"date-parts":[[2022,12,2]]},"references-count":18,"journal-issue":{"issue":"12","published-online":{"date-parts":[[2022,12]]}},"alternative-id":["axioms11120691"],"URL":"https:\/\/doi.org\/10.3390\/axioms11120691","relation":{},"ISSN":["2075-1680"],"issn-type":[{"type":"electronic","value":"2075-1680"}],"subject":[],"published":{"date-parts":[[2022,12,2]]}}}